After we had proved our theorems, we realized that one might derive Theorem REF  and Corollary REF  by combining results in [1]}, [2]}, [3]} and the Weyl–Kac character formula computations for affine Lie algebras in [4]}. However, we emphasize that our approach is more direct and does not rely on deep results from representation theory, in particular, Ariki's categorification theorem and Weyl–Kac character formula.
